We now define the system of PDEs driving the motion by curvature of networks; it remains to discuss the freedom in
the choice of the tangential component \(\zeta \)  in (REF ). At first sight, the choice of \(\zeta \)  in (REF )
does not seem important since it only corresponds to a reparametrization. However, observe that we cannot take \(\zeta ^{(j)} \equiv 0\) .
Indeed, following [1]}, if the solution is sufficiently smooth in space and time, then the values of \(\zeta ^{(j)}(t,0)\) 
are determined entirely by the values of \(\kappa ^{(j)}(t,0)\)  and vice versa. Different choices of \(\zeta ^{(j)}\)  taking these required values
at the vertices then do correspond to reparametrizations. However, suppose \(\zeta ^{(j)}\equiv 0\) , and let \(P(t)\)  denote the motion of any
interior vertex. Then for each \(j\) , \(P^{\prime }(t) = \partial _t \gamma ^{(j)}(t,0) = \kappa ^{(j)}(t,0) \nu ^{(j)}(t,0)\) ; since the \(\nu ^{(j)}\)  are independent,
we deduce that \(P^{\prime }(t) \equiv 0\) , i.e., the vertex remains fixed. However, imposing (REF ) and also requiring this vertex to be constant
leads to an overdetermined system of PDEs.
It is also the case that, due to the `gauge invariance', i.e., the possibility of reparametrization, (REF ) with \(\zeta \equiv 0\)  is not
strictly parabolic, see [2]}.
